Your favorite unit for each "class"

Myrrh.

Sword: Laegjarn
Lance: Catria
Axe: Helbindi, though Amelia is a close second due to investment I've given her.
Dragon: Myrrh / OG F!Corrin
Bow: Rebecca (4 unique pieces of Art!)
R!Tome: H!Nowi - S!Tana just isn't my thing...
B!Tome: Not Reinhardt...
G!Tome: F!Robin - someday I will properly build her
Dagger: H!Sakura
Staff: Veronica (I still wish she wasn't a staff user, but it wasn't my call)

Honorable Mention: Peri - She's crazy, but yet something about her...

I like Fliers overall, largely due to their mobility, which is also why I dislike Armors so much.

Favorite as in because of their power, or favorite as in character personality?

Overall: Lute, because Lute is Lute and Lute is superior after all.
Tomes: Lute, also Lilina (R), Linde (B), Nino (G).

Flying: Flying Lute when she gets on FEH, also BNinian and NYAzura because they are too op.

Armor: Armor Lute when she gets on FEH, then Effie because I love her and BEphraim and LYTiki and HMyrrh and all Hectors, Amelia...

Infantry: Lute, then Lute alt when she gets on FEH, also Innes, Eirika (OC).

Cavalry: Cavalry Lute when she is on FEH, also Gunnthra, Reinhard, Seth, SMEirika, LEirika, LEphraim...

Staff: Staff Lute when we get her on FEH, then Elise, Priscilla and Lucius. Ethlyn is also cute.

Dagger: Assasin Lute when she comes to FEH, then Jaffar. No one more.

Sword, axe, lance: Paladin Lute using all these weapons.
Did I mention Lute?

Sword: Alm
Lance: Shigure
Axe: L! Hector or Raven, can't pick
Bow: Innes
Red mage: Celica
Blue mage: Odin
Green mage: Soren
Staff: Lucius
Dragon: A. Tiki

Honorable mention goes to Prince Overdrive Marth

Edir: lol forgot dagger, Jaffar easily!

Sword: Masked Lucina
Lance: Ephraim
Axe: Legendary Hector (close 2nd Performing Azura)
Bow: Legendary Lucina
Dagger: Summer Linde (though I don't really use any Daggers frequently)
Dragon: Adrift Corrin (F)
Red Tome: Celica
Blue Tome: Ishtar (close 2nd/3rd Lute and Summer Corrin)
Green Tome: Lewyn (close 2nd Flying Nino)
Healer: Elise
Infantry: Masked Lucina
Cavalry: Legendary Ephraim
Flying: Summer Corrin
Armor: Legendary Hector

Guys with sexy body, so usually green axe units. Oh and Oliver.

Well, lets see
Blue cav mage Reinhardt
Red sword cav Xander
Green tome cav Cecilia

Blue tome infantry Lute
Green tome infantry Sonya/Boey
Red tome infantry Lyon with Owl+CC

Flying red tome Aversa/H!Nowi
Flying blue tome F!Morgan
Flying axe Michalis

I kinda love spell casters despite everything, also blade tomes in feh for its raw power (Morgan and Cecilia has blade+ on ). But having nice voice acting and good art is important to me.
I don't like staff users because you need to invest too much to get meh result. If not for razzle dazzle, healers don't have much to be good in all situation. Being locked out of great budget skills (blow, Lnd, seal skills to name a few), good supportive specials (other than primium new skills that Veronica and Loki have).
Same goes for daggers, they can be usefull (Saizo with smoke dagger and Felicia with her plate) but only thing daggers can do best is tank and debuff,but low atk and mt of daggers kills all offencive potential they have (excluding Sothe). There is woo dagger tho but it's seasonal, barb shuriken is nice so i might be wrong here.
Also i don't like armors for one space per turn thing, i know that armor march exist ( and i have it) but it still is not enough.But i like Hector, LA!Eliwood, Zephiel and H!Henry out of armors.

All depends on context and purpose :)

Offensive:
Sword: Karla
Axe: hard to decide between Cherche and Raven
Lance: Effie or Cordelia, or possibly Shigure
Dragon: M.Grima. RAWR bold fighter
Red tome: probably B.Tharja. I like OHKOs like Lilina a lot more after spiral though.
Blue tome: Ophelia no question. Reinhardt is pretty good too
Green tome: SF Nino probably
Bow: S.Takumi probably
Dagger: probably Sothe, but Felicia is pretty decent too. But sometimes I really do need just need more raw atk.
Staff: Veronica
Of all choices: Ophelia and Raven/Karla/Cherche. I prefer to hit res, and the more-or-less guaranteed AoE which ignores invisible buffs and weapon triangle is too strong. But she can't galeforce, so that's what the others are for.

Defensive:
Sword: Selena probably? Zephiel is pretty respectable too
Axe: Sheena easily, which is fantastic considering she's gen 1 isn't she?
Lance: probably Gwendolyn. If I wanted to dodge armoured weakness I'd just go for a manakete like Nowi
Dragon: probably F.Azurcorrin. Good spd, def, res. But hard to merge. If we're talking mergeable, M.Kana, or if no GHB/TT, Nowi.
Red tome: W.Tharja who are we kidding. Stats, stats, stats.
Blue tome: LA Lyn, same reason
Green tome: H.Henry, same reason
Bow: H.Jakob, or if no armours, then Faye. Among 3*-4* mergeable, Rebecca.
Dagger: Felicia or Saizo, depending on whether I want def or res.
Staff: Azama easily
Of all types: Sheena and M.Kana/F.dreamCorrin.

Balanced:
Sword: Soleil or Eirika
Axe: Libra probably
Lance: ....Fjorm? or maybe Sumia
Dragon: probably M.Kana, but either dream Corrin is pretty solid in this department too. Just that they're so hard to merge
Red tome: Camilla. :/ she has spd, good res, and access to flier buffs
Blue tome: Mae is the only one who comes to mind.
Green tome: B.Sanaki for same reason as dream Camilla
Bow: either Rebecca or someone like Leon/NY Corrin
Dagger: Felicia or Saizo, depending whether I want res or def. Same as my answer for defensive because daggers aren't generally that great at tanking.
Staff: if I'm trying to go halfway, probably Clarine or Nanna

Support:
Sword: Marth, and probably YT Olivia second. Marth is so versatile it's incredible.
Axe: NY Azura, because flier and hagoita and hone fliers. If not then definitely Camilla
Lance: B.Lucina
Dragon: Ninian
R tome: probably Aversa
G tome: PA Inigo probably. I don't find Soren's res high enough to ploy
B tome: B.Ninian who are we kidding
Bow: probably S.Noire actually. Fast, decent bulk, I would give her Clarisse's bow and null follow-up or firesweep and double smokes
Dagger: Legault. Smoke dagger, null follow-up
Staff: Veronica, but H.Mia is also a pretty decent option along with most other healers
Of all types: Veronica, generally speaking. But support units are more specific to the team comp than other types so even more than in other situations, depending on context.

Movement types:
Infantry: probably F.dreamCorrin or M.Kana, but Ophelia's a pretty strong contender.
Cavalry: Veronica
Flier: probably B.Ninian, but mainly because it's sweet to have such a flower on the team. For combat, I prefer YT Olivia who tanks better.
Armour: Sheena

In feh right ?
Dragon:myrrh
Sword:ayra
Bow:b lyn
Axe:don't have one,but we will see when Charlotte,jill or geese gets added to feh
Lance:b lucina
G tome:nino
B tome:ishtar
R tome:celica
Staff: maribelle

Here we go. Most of em are cuz of character.
I'm only counting original versions.

Sword: Eliwood (best noble lad)
Spear: Lukas (Nicest sociopath ever)
Axe: Michalis (Just enjoyably edgy. Axe doesn't have much competition)
Red Tome: Lyon (Sad lad)
Blue Tome: Mae (Ball of energy and fun)
Green Tome: Soren (Dunno why, I like him but I do not know why)
Bow: Innes (Biggest Tsundere)
Dagger:Jaffar (I'm a sucker for redemption)
Staff: Serra (So enjoyably stupid and selfish)
Dragonstone:Ninian (I am such a sucker for Eliwood x Ninian)

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A good list of honorable mentions:

Red: Arvis, Eldigan, Sigurd, Eirika, Flora, Gray, Joshua, Katarina (Very close second), Leo, Owain, Seth, Zephiel, BK/Zelgius

Blue: Berkut, Camus, Clair, Ephraim, Finn, Florina, L'Arachel, Odin, Olivier, Quan, The Rein, Shiro, Tana, Valter

Green: Bartre, Boey, Deirdre, Dorcas, Julia, Grima, Hawkeye, Hector, Legion, Lewyn, Minerva, Myrrh, Nino, Titania, Walhart

Colorless: Azama, Jacob, Jeorge, Kaze, Legault, Maribelle, Matthew, Sakura, Setsuna, Sothe, Takumi, Virion.

Too long? Probably

-Sword: Zelgius
He’s thicc, his special is awesome, he says the best lines, and he’s an upgrade from BK.
-Lance: Brave Hector
His weapon has QR5 and a slaying effect, and he has the best mixed defense I’ve seen in the game besides on L!Tiki and A!Corrin. His base kit is also godly and he’s such a dad.
-Axe: Brave Ephraim
His weapon has bold fighter, his own fighter skill is Guard but better, and he has the sexy attack superboon which puts him at an eye popping 60.
-Dragon: Adrift Female Corrin
She has the BST of a first gen armor unit and her mixed stats are the best in the game hands down, besides L!Tiki. Her base kit is amazing and her damage output reaches godly levels with triple bonds.
-Dagger: Halloween Kagero
She comes with Bold Fighter and can quad with some other daggers, but her base weapon is great and has Guard on it and inflicts massive debuffs. She has a wave C skill as well.
-Staff: Makoto
Her stats are part of a series of offensive oriented staffs, and he attack is sky high for staffs and probably the highest. Her weapon disables counter attacking, negating all damage done to your team on player phase (almost) and she can run the great dazzling/wrathful combo.
-Bow: Halloween Jakob
A powerhouse in the trust sense of the word and quit possibly he most terrifying unit in the game with Brave and bold build. Guaranteed quads with special activation. Enough said.

Sword - Owain
Lance - Ephraim / L! Ephraim
Axe - Raven (inb4 Gerik)
RedTome - Lyon
BlueTome - L'Arachel
GreenTome - Smolzura (she's cute)
Staff - Maria
Dagger - Legault
Bow - Innes

Honorable Mentions - Black Knight (Zelgius is cringy), Joshua, Marisa

Bow: Gordin (Everyone keeps picking the wrong green hair archer)
Dagger: H!Sakura
Lance: Abel
Sword: Roy
Axe: Cherche
R Tome: Lilina
B Tome: Kliff
G Tome: Sonya or LA!Lilina
Healer: Genny
Red Bow: Christmas Alm
Blue Bow: Christmas Gordin
Green Bow: Summer Klein

Sword:Legendary Ike/Legendary Eirika,hard to choose
Axe:Legendary Hector
Lance:Legendary Ephriam
Red Tome:Celica
Green Tome:Gunnthra
Blue Tome:Micaiah
Bow(in general):Legendary Lyn
Dagger (in general):Sothe
Staff:Mikoto
Dragon:Adrift Corrin

Honorable mentions go to-Zelgius

Sword: L!Ike
Lance: Fjorm
Axe: W!Lissa
R!Tome: HS!Micaiah
B!Toms: L'Arachel
G!Tome: B!Sanaki
Staff: Serra
Bow: L!Lyn
Dagger: P!Olivia
Dragon: Ninian

Fliers are my favorite of the type teams. Their mobility is really fun for kiting shenanigans.

Dragon - Nowi - Most dominant +10 merge for me
Bow - Brave Lyn - Destroys most things
Sword - BK - Gonna +10 him with Grails at some point
Lance - Hardin - Saving up to +10 him when he's a focus unit
Axe - Sheena - Latest +10 merge project
R tome - Aversa - Destroys most things
B tome - Reinhardt - Destroys most things
G tome - Cecilia - First 5 star +10 merge project, still good
Staff - Brave Veronica - Destroys most things and supports
Dagger - Sothe - Destroys well for the class, and supports.